R[adio]F[requency] filters based on thin-film bulk-acoustic-wave (BAW) resonators are being developed for applications such as mobile phones and wireless connectivity. The advantage of BAW technology is that devices are small, have good power handling (more than one Watt), cover the frequency range from about one to about twenty Gigahertz, and can exploit wafer-scale processing and packaging on silicon (Si).
Alternative technologies are ceramic electromagnetic (EM) wave filters and surface-acoustic-wave (SAW) filters. The former are relatively large and expensive for equivalent frequencies, while the latter require single-crystal materials such as lithium tantalate or quartz, are limited in practice to frequencies below about two Gigahertz, and also have limited power-handing capability.
A BAW resonator is essentially an acoustic cavity comprising a piezoelectric layer sandwiched between metal electrode layers. When an alternating electric signal is applied across these electrodes the energy is converted to mechanical form and a standing acoustic wave is excited. The principle mode of vibration in practical thin-film resonators is the fundamental thickness-extensional (TE1) acoustic mode, i. e. vibration is normal to the layers at a frequency for which half a wavelength of this mode is approximately equal to the total thickness of the cavity.
Two main types of resonator have been studied:
In the first of these resonator types, the film BAW resonator (=so-called FBAR; cf. T. W. Grudkowski, J. F. Black, T. M. Reeder, D. E. Cullen and R. A. Warner, “Fundamental mode VHF/UHF bulk acoustic wave resonators and filters on silicon”, Proc. IEEE Ultrasonics Symposium, pp. 829 to 833, 1980), a thin membrane 20 forms the cavity as shown in FIG. 1A (=side view of a membrane-based film BAW resonator) and in FIG. 1B (=top view of the FBAR of FIG. 1A).
Typical dimensions and materials of the FBAR (cf. FIGS. 1A and 1B) are about 300 micrometers for the thickness of the substrate 10, for example being made of silicon (Si), about hundred micrometers for the air gap 12 within the substrate 10, about one micrometer for the thickness of the thin membrane 20 (=etch-stop layer), for example being made of silicon dioxide (SiO2), about 100 micrometers for each top electrode 50 and the bottom electrode 30, and 1 μm for the respective pads 52 on the top electrodes 50 for example being made of aluminum (Al), and about three micrometers for the thickness of the C-axis normal piezoelectric layer 40, for example being made of aluminum nitride (AlN).
In the second of these resonator types, the solidly-mounted BAW resonator (=so-called SBAR; cf. K. M. Lakin, G. R. Kline and K. T. McCarron, “Development of miniature filters for wireless applications”, IEEE Trans. MTT-43, pp. 2933 to 2416, 1995) shown in FIG. 2A (=side view of a Bragg-reflector BAW resonator) and in FIG. 2B (=top view of the SBAR of FIG. 2A), the lower free surface of the membrane 20 (cf. FIG. 1A) is replaced by a set 20′ of acoustically mismatched layers 22, 24, which act to reflect the acoustic wave. This concept is analogous to the Bragg reflector in optics. The reflector layers 22, 24 are deposited on a solid substrate 10, typically glass or silicon (Si), so this structure is physically more robust than the FBAR.
Typical dimensions and materials of the SBAR (cf. FIG. 2A and 2B) are about 300 micrometers for the thickness of the substrate 10, for example being made of glass or silicon, about one micrometer for the thickness of each low mechanical impedance reflector layer 24, for example being made of silicon dioxide (SiO2), about one micrometer for the thickness of each high mechanical impedance reflector layer 22, for example being made of tungsten (W) or tantalum pentoxide (Ta2O5), with the set 20′ of acoustically mismatched layers 22, 24 being alternatingly formed of for example four high mechanical impedance reflector layers 22 and of for example four low mechanical impedance reflector layers 24 thus having a thickness of about eight micrometers, about 100 micrometers for each top electrode 50 and the bottom electrode 30, and 1 μm for the respective pads 52 on the top electrodes 50 for example being made of aluminum (Al), and about three micrometers for the thickness of the C-axis normal piezoelectric layer 40, for example being made of aluminum nitride (AlN).
Electrical connection to the bottom electrode may be through a via, as shown in FIG. 1. Alternatively, the via may be avoided by having the bottom electrode electrically floating, and forming two resonators in series, as shown in FIG. 2. With appropriate areas the two approaches are to first order electrically identical, and each may be used with either the FBAR or SBAR configuration.
A commonly used electrical equivalent circuit of a BAW resonator is shown in FIG. 3. C0, C1, L1 and R1 respectively characterize the static capacitance (=C0), the motional capacitance (=C1 ), the motional inductance (=L1) and the motional resistance (=R1) of the resonator itself, and together form the so-called Butterworth-Van Dyke BAW resonator model with added parasitics, i.e. the remaining components are electrical parasitics.
The three resistors characterize distinct types of energy loss: ohmic loss in the electrodes and interconnect (=Rs), loss due to stray electric fields in the substrate (=Rp), and mechanical losses associated with the resonance (=R1).
Dielectric loss is typically negligible.
The equivalent-circuit model according to FIG. 3 is useful for first-pass design of filters (and other circuits using BAW resonators).
A more physically based representation of a BAW resonator is the so-called Novotny-Benes BAW resonator model. This model provides a solution of the field equations in one dimension (1D). In this model it is assumed that the “resonator” as viewed in the direction normal to the layers is defined by the region of overlap between top electrode and bottom electrode. This will be referred to as the “internal” region, the space outside the edges being referred to as the “external” region.
If the configuration is as shown in the top view in FIG. 1B or in FIG. 2B, i.e. the top electrode area is substantially smaller than the bottom electrode, then the resonator edges coincide with the edges of the top electrode (except in the region of the interconnect), so the top electrode substantially defines the internal region (and vice versa when the bottom electrode is smaller).
In the 1D model it is effectively assumed that the mechanical fields and the electrical fields have significant spatial variations only in the x3-direction (i.e. direction normal to the layers) and are non-zero only in the internal region. All fields are assumed zero in the external region. Since the lateral dimensions of a typical resonator are much greater than layer thicknesses these are reasonable approximations in some respects.
The measured conductance G (=real part of resonator admittance Y) of a typical BAW resonator is compared over a wide band with predictions by both models in FIG. 4 (measurements: dashed line, Butterworth Van Dyke circuit model: solid line, !D Novotny Benes model: dotted line, anti-resonance fa). The level of agreement for the susceptance (=imaginary part of resonator admittance Y) is similar. The electrical parasitic components Cp, L5, Rp and R5 are included in both models.
Most, but crucially not all, features of the response are predicted by the 1D physical model. It is the behavior close to anti-resonance, which is not predicted by either of these models, that is central to this proposal. The additional effects seen in the response are associated with the true behavior of acoustic fields and of electric fields at the resonator edges.
The 1D model is itself non-physical in the sense that an abrupt change from non-zero acoustic field to zero acoustic field is only possible at surfaces adjacent to free space. However, there is no such interface at a plane normal to the layers at a resonator edge (except over a very small area at an edge of the top electrode as shown in FIG. 1A and in FIG. 2A). The piezoelectric layer and other layers are continuous. A more realistic model therefore requires the fields in such planes to be continuous.
The principle-missing contributions in the 1D model are the guided acoustic modes supported by the layer structure. Although a full model requires a three-dimensional (3D) field analysis, it is possible to understand the behavior at edges using a two-dimensional (2D) model, in which the acoustic fields and the electric fields are assumed to be non-uniform in both the x3-direction and the x1-direction (, i. e. a direction parallel to the layers).
In this 2D model the resonator edges are assumed to be in the planes x1=±W/2 where W is the resonator width. The x2-dimension of the resonator is, for the purposes of this analysis, assumed to be very great compared with the resonator width W, so fields are independent of x2. Such a 2D model can also be expected to give a good qualitative understanding of edge behavior even when the x1-dimension and the x2-dimension are more comparable. The advantage of the 2D model over a 3D numerical analysis is that it retains the analytical form, and therefore the physical insight, of the 1D model. The whole solution is a superposition of partial modes whose component fields have (in general complex) exponential x1- and x2-dependence.
The proposal is based on the understanding that the set of thin-film layers provides an acoustic waveguide, which allows guided acoustic modes to travel parallel to the layers. At any given frequency a number of such modes may exist, each mode n having a characteristic discrete x1-component of wave number kn. Waveguide mode solutions are found as combinations of partial modes in each layer by solving the coupled two-dimensional electrical and mechanical wave equations subject to appropriate boundary conditions at the layer surfaces.
The wave number kn may be real (indicating that the mode propagates unattenuated in the x1-direction), imaginary (indicating that the mode is attenuated in the x1-direction, i. e. the mode is a cut-off mode or evanescent mode) or complex (indicating that the mode propagates but with attenuation in the x1-direction).
The variation of the wave number kn with frequency is referred to as the dispersion relation for that mode. It is important in understanding. resonator edge behavior to recognize that the internal and external regions provide distinct waveguides. Although, in general, the same-guided mode types are supported in each waveguide, their dispersion relations are different.
FIG. 5 and FIG. 6 respectively show dispersion curves for the internal region (cf. FIG. 5) and for the external region (cf. FIG. 6) of a typical FBAR configuration. The convention in these diagrams is that frequency (measured in Gigahertz) is indicated on the y-axis, and the real and imaginary parts of normalized x1-component of wave number kn are indicated on the positive x-axis and negative x-axis respectively. (For convenience wave number kn is normalized at each frequency to that of an extensional wave propagating freely in the piezoelectric layer in the x3-direction.)
The layer thicknesses were chosen to give a fundamental thickness extensional (TE1) mode resonant frequency of two Gigahertz (as predicted by the 1D model). The shapes of the dispersion curves are independent of total thickness, provided the ratios of all layer thicknesses are kept constant.
Due to symmetry the negatives of the wave numbers kn shown are also solutions, and in the case of both real and imaginary parts of wave number kn being non-zero its complex conjugate (and negative of its complex conjugate) are also solutions. The lowest five modes are shown in each plot of FIG. 5 and of FIG. 6: flexural (F1), extensional (E1), fundamental thickness shear (TS1), fundamental thickness extensional (TE1) and second harmonic thickness shear (TE2).
It should be noted that the TE1 and TE2 branches form a continuous curve for both waveguide types. For the internal region their wave numbers kn are real above about 1.8 Gigahertz except over the interval from about 2 Gigahertz (TE1 mode cut-off) to about 2.1 Gigahertz (TE2 mode cut-oft) where one branch is imaginary.
Below 1.8 Gigahertz the two branches are complex conjugates. This indicates that the TE1 mode and the TE2 mode have similar field distributions and are likely to be strongly coupled at all frequencies. For the external region the two branches are complex conjugates below about 2.4 Gigahertz, and the two cut-off frequencies are off the diagram and therefore above 2.5 Gigahertz.
The other three modes (F1, E1 and TS1) have real wave numbers kn in both internal and external regions and therefore propagate unattenuated over the entire frequency range shown. All the higher modes (not shown) are strongly attenuated in this frequency range.
The condition of continuity of fields at resonator edges can only be satisfied by a linear superposition of the driving electro-acoustic field (, i.e. the 1D solution) and a combination of the guided modes in both internal and external regions. In principle, all guided modes in each region must be excited to some extent, since the continuity condition cannot otherwise be satisfied for all values of x3 in the edge plane.
In practice a few modes dominate. Here it is important to realize that the x3-dependence of the fields in each mode is a function of frequency, being close to the 1D solution for frequencies close to cut-off. Since the cut-off frequencies (particularly those of the dominant TE1 and TE2 modes) in the two regions differ considerably, the fields associated with nominally the same mode in the two regions also differ considerably.
Therefore a substantial contribution from other modes is needed to ensure continuity of the net field. This phenomenon is known as mode-conversion. One of its effects is that energy is lost through unattenuated propagation of the F1, E1 and TS1 modes away from the resonator.
Standing waves also occur due to guided modes excited at resonator edges traveling in opposite directions in the internal region. These standing waves are commonly referred to as inharmonic, because they are strongest at frequencies where an integer number of half-wavelengths of a guided mode corresponds approximately to the resonator width W.
The effect of edges on resonator admittance Y is therefore to introduce both loss which shows up as a contribution to conductance G (=real part of resonator admittance Y), due to guided modes scattered away from the resonator, and ripple in both real and imaginary parts of resonator admittance Y due to guided modes scattered back into the resonator. Although wave guiding and dispersion in an SBAR configuration is more complicated than in an FBAR configuration, similar arguments apply.
In the context of filter design the two frequencies of greatest interest in the response of a resonator are the resonance fr and the anti-resonance fa, the frequencies of its maximum and minimum admittance respectively. For high Q-factor resonances these are very close to the maximum and minimum of conductance. In the example whose response is shown in FIG. 4 these are at approximately 1.985 Gigahertz and 2.03 Gigahertz respectively.
FIG. 7 shows the very fine detail from FIG. 4 in the vicinity of the anti-resonance fa. This demonstrates the area of greatest disagreement between measurement and both the equivalent circuit and 1D simulation models. The measured response clearly shows the ripple and additional conductance (, i. e. loss) near the anti-resonance fa discussed above, which can only be explained by modeling behavior due to the edges.
All in all, it can be stated that acoustic energy escaping from the edges of resonators has been identified as one of the most significant sources of loss, and this occurs as a result of acoustic mode conversion at the physical discontinuity provided by the edge.